Some uniform effective results on André–Oort for sums of powers in Cⁿ

Abstract We prove an André–Oort-type result for a family of hypersurfaces in Cⁿ that is both uniform and effective. Let K_* denote the single exceptional imaginary quadratic field which occurs in the Siegel–Tatuzawa lower bound for the class number. We prove that, for m, n Z₀, there exists an effective constant c (m, n) 0 with the following property: if pairwise distinct singular moduli x₁, , xₙ with respective discriminants ₁, , ₙ are such that a₁ x₁ᵐ + + aₙ xₙᵐ Q for some a₁, , aₙ Q \0\ and \# \ ᵢ \;: \; {Q (ᵢ) = K_*\} 1, then ᵢ ᵢ c (m, n). In addition, we prove an unconditional and completely explicit version of this result when (m, n) = (1, 3) and thereby determine all the triples (x₁, x₂, x₃) of singular moduli such that a₁ x₁ + a₂ x₂ + a₃ x₃ Q for some a₁, a₂, a₃ Q \0\.